Global existence for the derivative NLS equation in the presence of solitons

TL;DR

This paper proves the global existence of solutions to the derivative nonlinear Schrödinger (DNLS) equation with initial data near solitons, using inverse scattering and Bäcklund transforms.

Contribution

It extends the inverse scattering method to include solitons for the DNLS equation, establishing global solutions in a broad function space.

Findings

Global solutions exist for initial data near solitons.

The method incorporates recent inverse scattering techniques.

Solitons are included in the solution framework.

Abstract

We prove the existence of global solutions to the DNLS equation with initial data in a large subset of $H^2(\mathbb R)\cap H^{1,1}(\mathbb R)$ containing a neighborhood of all solitons. We use the inverse scattering transform method, which was recently developed by D. Pelinovsky and Y. Shimabukuro, and an auto-B\"acklund transform in order to include solitons.